数值分析简明教程---课后答案

?(k?1)2(k)1(k)4?x2?x3??x1555?2(k)16(k)6?(k?1)，GG?Sx??x2?x3??2252525?6(k)321?(k?1)18(k)x?x?x3?2?3125125125?

??18?1，迭代收敛。 256.1 高斯消元法

1、（p.198，题2）用选列主元高斯消元法求解下列方程组：

?x1?x2?x3??4?（1）?5x1?4x2?3x3??12

?2x?x?x?1123?1?2x1?3x2?5x3?5?（2）?3x1?4x2?7x3?6

?x?3x?3x?523?15?4?1?11?4??5?43?12??1r1?r2????r1?r2??51【解】 （1）?5?43?12???1?11?4???0?5??21111??21111???????21

3?12?28??? 55?111?????5?43?12??2r1?r3?5?43?12??5?43?12?5?r3??5???5?r3???0?12?8???0?12?8???0?12?8?

13179???21111???013?1790??????555??????5?43?12?1r2?r3?5?43?12?13r3?5?43?12?r2?r3??13???25???013?179???013?179???013?179?

2525???0?12?8???001?100??????1313???x?794x?3x3?124?6?3?(?1)?12?6,x1?2??3. 所以： x3??1，x2?31355?2355??3476??2r1?r2?3476??3476????r1?r2??3?11?3r2?（2）?3476???2355???01???0113?

33??1335??1335??1335???1335?????????1?r1?r33??3??0??0?41537123??3476?6??3476?3r3?r2?r3????3??0113???0529? ???0113?3??0529?????21 / 23

?3476?5r3?3476???3????0529???0529?

?003565??0012??????2x3?9?4x2?7x3?6?4?1?7?2?6所以： x3?2，x2??1,x1????4.

5351?r2?r35 2、（p.199，题9）计算下列三阶坡度阵的条件数：

??111?2（1）??113?1??234?。 ?111????345????111?213?【解】令：A???11??2314?，先求A-1。 ?11????345??

??111100??23100???11??111?1r1?r2?123?234010??2???011?1210?? ?111??1212???345001??111????345001????111?12?1?r2?23100???1?1r1?r2?1?011323100??111?6120?????0111?6120?

4???345001??????01245?1301?????112r2?r3?111100??111???0213?1?180r3??23100??1?6120??1?16120???01?130?180180???001806?11???00??????111?1?r?3?r2??23100???1r3?r1?10?960?010?36192?180?3??0210?36??00130?180180???192?????00130?180?22 / 23

?60???180?180? ???

1?r2?r129?3630??3630??100?9?，所以 A?1???36192?180?

??010?36192?180???????00130?180180???30?180180???1?A?

最后求得条件数为：cond(A)?A

??11?408?748 623 / 23